Active networks having biquadratic transfer functions

ABSTRACT

First, second, and third single-input, single-output operational amplifiers having associated input resistors and feedback resistors are connected in series between input and output terminals. First and second capacitors are also connected across associated first and second amplifiers to provide a second-order network transfer function. A fourth feedback resistor is connected between the output terminal of the third amplifier and the amplifier side of the first input resistor. The amplifier input sides of the second and third input resistors are also connected through associated feed-forward resistors to the one side of the first input resistor that is spaced from the first amplifier. Appropriate selection of element values enables synthesis of circuits having prescribed biquadratic transfer functions.

BACKGROUND OF INVENTION

This invention relates to active networks and more particularly tonetworks having biquadratic transfer functions.

A biquad circuit is considered to be one having a biquadratic transferfunction which is generally defined as one where the denominatorincludes a second-order or s² term and the numerator is of degree two orless. Biquad circuits employing three and four operational amplifiers tosimulate circuits having prescribed biquadratic transfer functions aredescribed in prior-art literature such as "A Step-by-Step Active-FilterDesign" by J. Tow, IEEE Spectrum, December 1969, pp. 64 - 68; and theBiquad -- Parts I and II, by Lee C. Thomas, IEEE Transactions on CircuitTheory, volume Ct 18-3, May 1971, pp. 350-361.

An object of this invention is the provision of a versatile networkhaving only three active elements and which can be adjusted to haveprescribed second-order transfer functions.

DESCRIPTION OF PREFERRED EMBODIMENTS

This invention will now be described in relation to the single FIGURE ofdrawing, which is a schematic circuit diagram of a network 55 embodyingthis invention. The network 55 here comprises three amplifiers A1, A2,and A3 and associated input resistors 1, 2, and 3 which are connected inseries between line 5 of an input port E_(in) and line 6 of an outputport E_(out). The amplifiers are shown as having single-ended inputs andoutputs. Alternatively, the amplifiers may have differential inputs anddifferential outputs, the other input lines being connected to groundand only one of the output lines being used. The actual polarity of theamplifier connections is selected to ensure that the network is DCstable. Terminals of the amplifiers are also connected to supply DC biasvoltages in the conventional manner. The amplifiers A1 - A3 arepreferably operational amplifiers. They are considered in the followingdiscussion to be ideal amplifiers with zero output impedance, infiniteinput impedance, and infinite voltage gain. Although such an idealoperational amplifier does not exist in practice, this does notseriously affect the operation of the network 55. A non-ideal amplifiermerely introduces extraneous terms in the transfer function for thenetwork, which can easily be compensated for in a manner well known inthe art, in order to produce a desired transfer function. In practice,the voltage gains K of the amplifiers are in the order of 100 or more.The values of the gains K of the amplifiers are positive so that theresultant term -K is negative to provide inversion of the input signaland to ensure that the amplifiers are stable.

Feedback resistors 11, 12, and 13 are connected across associatedamplifiers A1, A2, and A3. The resistances of feedback resistors 11 and12 are designed so that the associated feedback amplifiers A1 and A2have low DC gains, in the order of ten. This makes it possible to employamplifiers in this network which have lower gain bandwidth products,e.g., in the order of ten, then might be thought possible. This network55 employs only two capacitors 15 and 16, which is the theoreticalminimum number of capacitors for a network having a quadratic transferfunction. The capacitors 15 and 16 are part of the feedback impedancesconnected across associated amplifiers A1 and A2 to provideintegrator-type configurations there which are desirable for stabilityat high frequencies. The capacitances of elements 15 and 16 arepreferably the same values for simplicity of construction, although thisis not essential. The design of the network is also simplified if thefeedback resistors 11 and 12 have the same values of resistance,although they may be different.

The third amplifier A3 is an inverting amplifier having the feedbackresistor 13 connected across it. The resistances of the pairs ofelements 3 and 13 are also preferably the same values for simplifyingthe design of the inverting amplifier A3, although this is notessential. Feed-forward resistors 19 and 20 are also connected betweeninput line 5 and the amplifier-input sides of resistors 2 and 3,respectively. Finally, a feedback resistor 21 is connected between theoutput line 6 from A3 and the input line 24 to A1.

The transfer function of a general biquadratic equation is representableas ##EQU1## where s is the complex frequency variable, and n2, n1, n0,d2, d1, and d0 are positive real coefficients.

Assume, for the sake of simplicity of illustration, that the twoelements of each pair of resistors 2 and 21, 3 and 13, and 11 and 12,and of capacitors 15 and 16 have the same values; that the values ofnetwork elements are at least initially normalized with respect to thecapacitance of the capacitors 15 and 16; and that the values ofresistors 3 and 13 are selected to make the normalized values thereofalso be unity. Elementary network analysis for this circuitconfiguration, with ideal amplifiers and the above assumptions, thengives a network transfer function of ##EQU2## where the g's representthe normalized conductances of associated resistors as indicated in thefigure. Equation (2) is the general biquadratic transfer function inequation (1), expressed in terms of values of elements in network 55. Byspecifying only the normalized conductances g1 to g5 of associatedresistors, this network can be tailored to have different prescribedsecond-order transfer functions. This reduces the requirements on valuesof the capacitors 15 and 16.

Requiring that values of elements satisfy the relationships ##EQU3##then equation (2) reduces to where n0, n2, d0, and d1 are the generalcoefficients specified in equation (3), d2 = 1 and n1 = 0 here. Thetransfer function in equation (4) is a biquadratic equation, of course.More specifically, equation (4) is the transfer function for either alowpass or a highpass notch filter. By selecting a particular transferfunction to be satisfied by the network, the frequency response of thenetwork is specified and particular values of the biquadraticcoefficients are obtained. These coefficient values are employed in therelationships in equation (3) to provide specific values of thenormalized conductances there. The circuit is then denormalized bysetting the capacitances equal to a particular desired capacitance C_(o)and the resistances equal to 1/(gi 2π f_(o) C_(o)), where i = 1, 2, . .. 5, and f_(o) is the frequency at which the filter is desired tooperate. In this instance, both of the resistors 2 and 21 have the samevalues as do the elements of the pairs of resistors 3 and 13, ofresistors 11 and 12, and of capacitors 15 and 16. Alternatively, thecircuit may be denormalized by denormalizing each amplifier section byitself to provide a network in which the elements of an element pairhave different values. In this latter case, the representation of thetransfer function of network 55 is then more complex than that shown inequation (4), although the network 55 in both cases has the samefrequency response.

The same network 55 is made to simulate an all-pass filter structurehaving a transfer function ##EQU4## by requiring that element valuessatisfy the relationships ##EQU5## where n1 and n0here are quadraticcoefficients which have positive values, A is the gain of the circuit55, d2 = n2 = 1, d1 = n1, and d0 = n0. Such a network has a constantloss and a variable phase delay. The particular frequencycharacteristics of this structure defined by equation (5) are set byselecting specific values of the remaining coefficients n1 and n0 andthe gain A, and then denormalizing element values as was generallyoutlined above. The same network is caused to simulate an attenuationequalizer by changing the values of only g2 and g4 such that theysatisfy the relationships

    g2 = d1/2 and g4 = g5(d1 - n1).                            (7)

The transfer function for this attenuation equalizer is representable as##EQU6## where d1, n1, and n0 here are quadratic coefficients, d2 = n2 =1, d0 = n0, and d1 is greater than n1. In summary, the values of networkelements for this attenuation equalizer satisfy the relationship##EQU7## Further, the aforementioned network defined by equation (8)simulates an attenuation and phase equalizer satisfying the generalizedtransfer function ##EQU8## by merely requiring that the coefficient n1be negative. As stated previously, particular values of the quadraticcoefficients, and finally element values, are obtained afer firstselecting a particular transfer function to be satisfied by the network.

In an alternate embodiment of the invention, the network 55 simulates ahighpass, all-pole filter by requiring that the normalized elementvalues satisfy the relationship ##EQU9## This highpass circuit has thegeneralized transfer function ##EQU10## where d2 = 1 and n1 = n0 = 0.

This same network 55 is modified to operate as a bandpass filter definedby the generalized transfer function ##EQU11## where d2 = 1 and n2 = n0= 0, by requiring that the element values satisfy the relationships

    g1 = g2 g4/g3; g2 = d1/2; g3 = √d0 - g2.sup.2 ; g4 = n1; g5 = 0. (14)

In this bandpass circuit, the resistor 20 has an infinite impedance suchthat it is replaced by an open circuit. The same network 55 is modifiedto operate as a lowpass, all-pole filter having a general transferfunction ##EQU12## where d2 = 1 and n2 = n1 = 0, by requiring that theelement values satisfy the relationships

    g1 = n0/g3; g2 = d1/2; g3 = √d0 - g2.sup.2 ; g4 = g5 = 0. (16)

In this lowpass circuit, the resistances of both of the feedforwardresistors 19 and 20 are infinite such that they are replaced by opencircuits.

The network 55 may also be made to simulate a circuit having thetransfer function ##EQU13## where d2 = n2 = 1 and d1 = n1 = 0, byrequiring that the element values satisfy the relationships ##EQU14## Inthis circuit, the resistors 11, 12, and 19 are replaced by opencircuits. Such a filter circuit has particular application in networkshaving multiple feedback paths.

By way of example, a typical highpass notch filter has a transferfunction of the form in equation (4) where d0 = 1, d1 = 0.1, n0 = 0.25,and n2 = 1. This is a highpass filter with a normalized loss pole at ω =0.5. Values of normalized conductances computed from equation (3) are g1= 0.2528, g2 = 0.05, g3 = 0.9987, g4 = 0.1, and g5= 1. If the filter isto operate at 1 kHz, the capacitances of capacitors 15 and 16 may be 10nanofarads. Denormalized resistances of associated resistors aretherefore R1 = 62.96 kΩ; R2 = R21 = 15.92 kΩ; R3 = R13 = 15.92 kΩ; R11 =R12 = 318.3 kΩ; R19 = 159.2 kΩ; and R20 = 15.92 kΩ.

Although this invention is described in relation to preferredembodiments thereof, variations and modifications will be apparent tothose skilled in the art. By way of example, the output of the networkmay be coupled from either of the low-impedance output lines 25 and 26at the amplifiers, as well as from the low-impedance output line 6,although the transfer function of the resulting network will then bedifferent from those specified above.

What is claimed is:
 1. A frequency-selective network satisfying aprescribed transfer function, the network including input and outputports with only one terminals thereof being grounded and having atransfer function which is generally representable as ##EQU15## wheren2, n1, n0, d2, d1, and d0 are constant coefficients which may be zeroand s is the complex frequency variable, the network comprising: first,second, and third operational amplifiers each having at least one inputline and having at least one output line; first, second, and third inputresistors in the one input lines of associated amplifiers; said first,second, and third resistors and first, second, and third amplifiersbeing electrically connected in series, the one input line of said firstamplifier being electrically connected through said first resistor tothe other -- ungrounded input terminal; the parallel combination of afourth resistor and a first capacitor electrically connected between theone output line and the one input line of said first amplifier; theparallel combination of a fifth resistor and a second capacitorelectrically connected between the one output line and the one inputline of said second amplifier; a sixth resistor electrically connectedbetween the one output line and the one input line of said thirdamplifier; a seventh resistor electrically connected between the thirdamplifier one output line and the first amplifier one input line; aneighth resistor electrically connected between the second amplifier oneinput line and the other -- ungrounded input terminal; a ninth resistorelectrically connected between the third amplifier one input line andthe other -- ungrounded input terminal; and means for coupling the other-- ungrounded output terminal to one output line of one of saidamplifiers; the network transfer function being a function of thenormalized conductances g1, g2, g3, g4, and g5 of said first, fourth,second, eighth, and ninth resistors, respectively, being prescribed byselected values of these conductances which are at least initiallynormalized with respect to the capacitances of said first and secondcapacitors with the normalized conductances of said third and sixthresistors being at least initially set to unity, and being representableas ##EQU16## normalized capacitances of said first and second capacitorsboth being unity; normalized conductances of said fourth and fifthresistors being the same values; normalized conductances of said secondand seventh resistors being the same values; and normalized conductancesof said third and sixth resistors being the same values.
 2. Afrequency-selective network satisfying a prescribed transfer functionfor operating as a notch filter, the network including input and outputports with only one terminals thereof being grounded and having atransfer function which is generally representable as ##EQU17## wheren2, n1, n0, d2, d1, and d0 are constant coefficients which may be zeroand s is the complex frequency variable, the network comprising: first,second, and third operational amplifiers each having at least one inputline and having at least one output line; first, second, and third inputresistors in the one input lines of associated amplifiers; said first,second, and third resistors and first, second, and third amplifiersbeing electrically connected in series, the one input line of said firstamplifier being electrically connected through said first resistor tothe other - ungrounded input terminal; the parallel combination of afourth resistor and a first capacitor electrically connected between theone output line and the one input line of said first amplifier; theparallel combination of a fifth resistor and a second capacitorelectrically connected between the one output line and the one inputline of said second amplifier; a sixth resistor electrically connectedbetween the one output line and the one input line of said thirdamplifier; a seventh resistor electrically connected between the thirdamplifier one output line and the first amplifier one input line; aneighth resistor electrically connected between the second amplifier oneinput line and the other -- ungrounded input terminal; a ninth resistorelectrically connected between the third amplifier one input line andthe other -- ungrounded input terminal; and means for coupling the other-- ungrounded output terminal to one output line of one of saidamplifiers; the network transfer function being a function of thenormalized conductances g1, g2, g3, g4, and g5 of said first, fourth,second, eighth, and ninth resistors, respectively, being prescribed byselected values of these conductances which are at least initiallynormalized with respect to the capacitances of said first and secondcapacitors with the normalized conductances of said third and sixthresistors being at least initially set to unity, and being representableas ##EQU18## the coefficients and normalized conductances of saidresistors satisfying the requirements that ##EQU19## the prescribedtransfer function being representable as ##EQU20## where n1 = 0 and d2=
 1. 3. A frequency-selective network satisfying a prescribed transferfunction for operating as an all-pass filter, the network includinginput and output ports with only one terminals thereof being groundedand having a transfer function which is generally representable as##EQU21## where n2, n1, n0, d2, d1, and d0 are constant coefficientswhich may be zero and s is the complex frequency variable, the networkcomprising: first, second, and third operational amplifiers each havingat least one input line and having at least one output line; first,second, and third input resistors in the one input lines of associatedamplifiers; said first, second, and third resistors and first, second,and third amplifiers being electrically connected in series, the oneinput line of said first amplifier being electrically connected throughsaid first resistor to the other -- ungrounded input terminal; theparallel combination of a fourth resistor and a first capacitorelectrically connected between the one output line and the one inputline of said first amplifier; the parallel combination of a fifthresistor and a second capacitor electrically connected between the oneoutput line and the one input line of said second amplifier; a sixthresistor electrically connected between the one output line and the oneinput line of said third amplifier; a seventh resistor electricallyconnected between the third amplifier one output line and the firstamplifier one input line; an eighth resistor electrically connectedbetween the second amplifier one input line and the other -- ungroundedinput terminal; a ninth resistor electrically connected between thethird amplifier one input line and the other -- ungrounded inputterminal; and means for coupling the other -- ungrounded output terminalto one output line of one of said amplifiers; the network transferfunction being a function of the normalized conductances g1, g2, g3, g4,and g5 of said first, fourth, second, eighth, and ninth resistors,respectively, being prescribed by selected values of these conductanceswhich are at least initially normalized with respect to the capacitancesof said first and second capacitors with the normalized conductances ofsaid third and sixth resistors being at least initally set to unity, andbeing representable as ##EQU22## the coefficients and normalizedconductances of said resistors satisfying the requirements that##EQU23## where A is the gain of the network, the prescribed transferfunction being presentable as ##EQU24## where d2 = n2 = 1, d1 = n1, andd0 = n0.
 4. A frequency -selective network satisfying a prescribedtransfer function for operating as an attenuation equalizer, the networkincluding input and output ports with only one terminals thereof beinggrounded and having a transfer function which is generally representableas ##EQU25## where n2, n1, n0, d2, d1, and d0 are constant coefficientswhich may be zero and s is the complex frequency variable, the networkcomprising: first, second, and third operational amplifiers each havingat least one input line and having at least one output line; first,second, and third input resistors in the one input lines of associatedamplifiers; said first, second, and third resistors and first, second,and third amplifiers being electrically connected in series, the oneinput line of said first amplifier being electrically connected throughsaid first resistor to the other -- ungrounded input terminal; theparallel combination of a fourth resistor and a first capacitorelectrically connected between the one output line and the one inputline of said first amplifier; the parallel combination of a fifthresistor and a second capacitor electrically connected between the oneoutput line and the one input line of said second amplifier; a sixthresistor electrically connected between the one output line and the oneinput line of said third amplifier; a seventh resistor electricallyconnected between the third amplifier one output line and the firstamplifier one input line; an eighth resistor electrically connectedbetween the second amplifier one input line and the other -- ungroundedinput terminal; a ninth resistor electrically connected between thethird amplifier one input line and the other -- ungrounded inputterminal; and means for coupling the other -- ungrounded output terminalto one output line of one of said amplifiers; the network transferfunction being a function of the normalized conductances g1, g2, g3, g4,and g5 of said first, fourth, second, eighth, and ninth resistors,respectively, being prescribed by selected values of these conductanceswhich are at least initially normalized with respect to the capacitancesof said first and second capacitors with the normalized conductances ofsaid third and sixth resistors being at least initially set to unity,and being representable as ##EQU26## the coefficients and normalizedconductances of said resistors satisfying the requirements that##EQU27## the prescribed transfer function being representable as##EQU28## where d2 = n2 = 1, d1 is greater than n1, and d0 = n0.
 5. Thenetwork according to claim 4, wherein the value of n1 is negative, thenetwork now operating as an attenuation and phase equalizer.
 6. Afrequency-selective network satisfying a prescribed transfer functionfor operating as a high-pass, all-pole filter structure, the networkincluding input and output ports with only one terminals thereof beinggrounded and having a transfer function which is generally representableas ##EQU29## where n2, n1, n0, d2, d1, and d0 are constant coefficientswhich may be zero and s is the complex frequency variable, the networkcomprising: first, second, and third operational amplifiers each havingat least one input line and having at least one output line; first,second, and third input resistors in the one input lines of associatedamplifiers; said first, second, and third resistors and first, second,and third amplifiers being electrically connected in series, the oneinput line of said first amplifier being electrically connected throughsaid first resistor to the other -- ungrounded input terminal; theparallel combination of a fourth resistor and a first capacitorelectrically connected between the one output line and the one inputline of said first amplifier; the parallel combination of a fifthresistor and a second capacitor electrically connected between the oneoutput line and the one input line of said second amplifier; a sixthresistor electrically connected between the one output line and the oneinput line of said third amplifier; a seventh resistor electricallyconnected between the third amplifier one output line and the firstamplifier one input line; an eighth resistor electrically connectedbetween the second amplifier one input line and the other -- ungroundedinput terminal; a ninth resistor electrically connected between thethird amplifier one input line and the other -- ungrounded inputterminal; and means for coupling the other -- ungrounded output terminalto one output line of one of said amplifiers; the prescribed transferfunction being a function of the normalized conductances g1, g2, g3, g4,and g5 of said first, fourth, second, eighth, and ninth resistors,respectively, being prescribed by selected values of these conductanceswhich are at least initially normalized with respect to the capacitancesof said first and second capacitors with the normalized conductances ofsaid third and sixth resistors being at least initially set to unity,and being representable as ##EQU30## where n1 = n0 = 0 and d2 = 1,values of other coefficients and normalized conductances of saidresistors satisfying the relationships ##EQU31##
 7. Afrequency-selective network satisfying a prescribed transfer functionfor operating as a bandpass filter, the network including input andoutput ports with only one terminals thereof being grounded and having atransfer function which is generally representable as ##EQU32## wheren2, n1, n0, d2, d1, and d0 are constant coefficients which may be zeroand s is the complex frequency variable, the network comprising: first,second, and third operational amplifiers each having at least one inputline and having at least one output line; first, second, and third inputresistors in the one input lines of associated amplifiers; said first,second, and third resistors and first, second, and third amplifiersbeing electrically connected in series, the one input line of said firstamplifier being electrically connected through said first resistor tothe other -- ungrounded input terminal; the parallel combination of afourth resistor and a first capacitor electrically connected between theone output line and the one input line of said first amplifier; theparallel combination of a fifth resistor and a second capacitorelectrically connected between the one output line and the one inputline of said second amplifier; a sixth resistor electrically connectedbetween the one output line and the one input line of said thirdamplifier; a seventh resistor electrically connected between the thirdamplifier one output line and the first amplifier one input line; aneighth resistor electrically connected between the second amplifier oneinput line and the other -- ungrounded input terminal; and means forcoupling the other -- ungrounded output terminal to one output line ofone of said amplifiers; the prescribed transfer function being afunction of the normalized conductances g1, g2, g3 and g4 of said first,fourth, second, and eighth resistors, respectively, being prescribed byselected values of these conductances which are at least initiallynormalized with respect to the capacitances of said first and secondcapacitors with the normalized conductances of said third and sixthresistors being at least initially set to unity, and being representableas ##EQU33## where n2 = n0 = 0 and d2 = 1, values of other coefficientsand normalized conductances of said resistors satisfying therelationships g1 = g2g4/g3; g2 = d1/2; g3 = √d0 - g2² ;g4 = n1.
 8. Afrequency-selective network satisfying a prescribed transfer functionfor operating as a low-pass, all-pole filter, the network includinginput and output ports with only one terminals thereof being groundedand having a transfer function which is generally representable as##EQU34## where n2, n1, n0, d2, d1, and d0 are constant coefficientswhich may be zero and s is the complex frequency variable, the networkcomprising: first, second, and third operational amplifiers each havingat least one input line and having a least one output line; first,second, and third input resistors in the one input lines of associatedamplifiers; said first, second, and third resistors and first, second,and third amplifiers being electrically connected in series, the oneinput line of said first amplifier being electrically connected throughsaid first resistor to the other -- ungrounded input terminal; theparallel combination of a fourth resistor and a first capacitorelectrically connected between the one output line and the one inputline of said first amplifier; the parallel combination of a fifthresistor and a second capacitor electrically connected between the oneoutput line and the one input line of said second amplifier; a sixthresistor electrically connected between the one output line and the oneinput line of said third amplifier; a seventh resistor electricallyconnected between the third amplifier one output line and the firstamplifier one input line; and means for coupling the other -- ungroundedoutput terminal to one output line of one of said amplifiers; theprescribed transfer function being a function of the normalizedconductances g1, g2, and g3 of said first, fourth, and second resistors,repectively, being prescribed by selected values of these conductanceswhich are at least initially normalized with respect to the capacitancesof said first and second capacitors with the normalized conductances ofsaid third and sixth resistors being at least initially set to unity,and being representable as ##EQU35## where n2 = n1 = 0 and d2 = 1,values of the other coefficients and the normalized conductances of saidresistors satisfying the relationships

    g1 = n0/g3; g2 = d1/2;g3= √d0 - g2.sup.2.


9. A frequency-selective network satisfying a prescribed transferfunction, the network including input and output ports with only oneterminals thereof being grounded and having a transfer function which isgenerally representable as ##EQU36## where n2, n1, n0, d2, d1, and d0are constant coefficients which may be zero and s is the complexfrequency variable, the network comprising: first, second, and thirdoperational amplifiers each having at least one input line and having atleast one output line; first, second, and third input resistors in theone input lines of associated amplifiers; said first, second, and thirdresistors and first, second, and third amplifiers being electricallyconnected in series, the one input line of said first amplifier beingelectrically connected through said first resistor to the other --ungrounded input terminal; a first capacitor electrically connectedbetween the one output line and the one input line of said firstamplifier; a second capacitor electrically connected between the oneoutput line and the one input line of said second amplifier; a fourthresistor electrically connected between the one output line and the oneinput line of said third amplifier; a fifth resistor electricallyconnected between the third amplifier one output line and the firstamplifier one input line; a sixth resistor electrically connectedbetween the third amplifier one input line and the other -- ungroundedinput terminal; and means for coupling the other -- ungrounded outputterminal to one output line of one of said amplifiers; the particulartransfer function being a function of the normalized conductances g1,g2, and g3 of said first, second and sixth resistors, respectively,being prescribed by selected values of these conductances which are atleast initially normalized with respect to the capacitances of saidfirst and second capacitors with the normalized conductances of saidthird and fourth resistors being at least initially set to unity, andbeing representable as ##EQU37## where A is the gain of network, n2 = d2= 1, and n1 = d1 = 0, values of other coefficients and the normalizedconductances of said resistors satisfying the relationships ##EQU38##10. A frequency-selective network satisfying a prescribed transferfunction for operating as an attenuation and phase equalizer, thenetwork including input and output ports with only one terminals thereofbeing grounded and having a transfer function which is generallyrepresentable as ##EQU39## where n2, n1, n0, d2, d1, and d0 are constantcoefficients which may be zero and s is the complex frequency variable,the network comprising: first, second, and third operational amplifierseach having at least one input line and having at least one output line;first, second, and third input resistors in the one input lines ofassociated amplifiers; said first, second, and third resistors andfirst, second, and third amplifiers being electrically connected inseries, the one input line of said first amplifier being electricallyconnected through said first resistor to the other -- ungrounded inputterminal; the parallel combination of a fourth resistor and a firstcapacitor electrically connected between the one output line and the oneinput line of said first amplifier; the parallel combination of a fifthresistor and a second capacitor electrically connected between the oneoutput line and the one input line of said second amplifier; a sixthresistor electrically connected between the one output line and the oneinput line of said third amplifier; a seventh resistor electricallyconnected between the third amplifier one output line and the firstamplifier one input line; an eighth resistor electrically connectedbetween the second amplifier one input line and the other -- ungroundedinput terminal; a ninth resistor electrically connected between thethird amplifier one input line and the other -- ungrounded inputterminal; and means for coupling the other -- ungrounded output terminalto one output line of one of said amplifiers; the network transferfunction being a function of the normalized conductances g1, g2, g3, g4and g5 of said first, fourth, second, eighth, and ninth resistors,respectively, being prescribed by selected values of these conductanceswhich are at least initially normalized with respect to the capacitancesof said first and second capacitors with the normalized conductances ofsaid third and sixth resistors being at least initially set to unity,and being representable as ##EQU40## the coefficients and normalizedconductances of said resistors satisfying the requirements that##EQU41## the transfer function of the network being representable as##EQU42## where A is the gain of the network, d2 = n2 = 1 and d0 = n0.